Response

Resurgence affects only creatures you control at the time it resolves. Creatures that come under your control later in the turn won't gain first strike and vigilance.

If you cast Resurgence when it's not a main phase (perhaps with Omnispell Adept's ability), creatures you control gain first strike and vigilance, but there's no additional combat phase or main phase. If you cast it during an opponent's main phase, there are additional combat and main phases, but that player attacks during that combat phase, not you.

If you copy a spell that's half of a split card, the copy copies that same half. For example, if you copy Assault, the copy is also Assault, not Battery.